Information Geometry of Complex Hamiltonians and Exceptional Points

Information geometry provides a tool to systematically investigate parameter sensitivity of the state of a system. If a physical system is described by a linear combination of eigenstates of a complex (that is, non-Hermitian) Hamiltonian, then there can be phase transitions where dynamical properties of the system change abruptly. In the vicinities of the transition points, the state of the system becomes highly sensitive to the changes of the parameters in the Hamiltonian. The parameter sensitivity can then be measured in terms of the Fisher-Rao metric and the associated curvature of the parameter-space manifold. A general scheme for the geometric study of parameter-space manifolds of eigenstates of complex Hamiltonians is outlined here, leading to generic expressions for the metric.


12
In statistical physics, if a system is in equilibrium with a heat bath at inverse temperature β, then the 13 state of the system is characterised by the canonical phase-space density function where H(x) is the Hamiltonian function on phase space Ω and the partition function Z(β) is given 15 by the integral of the Boltzmann weight exp(−βH) over Ω . Systems having sufficiently rich inter- 16 particle interactions can exhibit phase transitions. Typically a phase transition is associated with the If the state of a system is described by a density function (or a discrete set of probabilities) that 23 lacks analyticity in the first place, however, then a phase transition can be seen without involving the 24 mathematically cumbersome operation of thermodynamic limit. Such situations arise in many physical 25 contexts. For example, if an isolated quantum system is in a microcanonical state having support on the 26 level surface of the expectation of the Hamiltonian, then the density of states is not analytic and one can 27 find thermal phase transitions in small quantum systems [1].  at the exceptional point. The result is also compared to the perturbative analysis of §6. 75 We remark that while physical effects associated with the existence of exceptional points have been smooth curve on the unit sphere in H. In particular, in the limit β → ∞ the equilibrium state |ξ(β) of the system approaches the 'ground state' of the Hamiltonian, i.e. a state with minimum energy. (Note that unless the parameter is changed adiabatically, the physical state of the system will not traverse the path 98 |ξ(β) since a rapid change of temperature momentarily brings the state of the system out of equilibrium.

99
Hence we are not concerned here with the out-of-equilibrium dynamics of the system as such. Rather, 100 we are interested in how an equilibrium configuration at one temperature is related to the equilibrium 101 configuration at another temperature, and this is characterised by the path |ξ(β) . Our analysis thus infinite-dimensional unit sphere offers a visual characterisation of the situation.

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In the case of a one-parameter family of states |ξ(β) the parametric sensitivity can be measured 110 in terms of the squared 'velocity' (metric) and the squared 'acceleration' (curvature) of the curve. By 111 squared velocity, which we shall denote by G, we mean the inner product where the dot represents differentiation with respect to β, and the factor of four is purely conventional where we have written β = 1/k B T , with k B the Boltzmann constant.

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Similarly, the acceleration vector |α(β) of the curve is defined by where |ξ(β) = ∂ 2 β |ξ(β) . In terms of the acceleration vector the intrinsic curvature K 2 of the curve 123 |ξ(β) is given by In the case of the canonical state (1) a calculation shows [28] that where we have written ∆H k to mean the k th central moment of the Hamiltonian in the thermal state (1).
More generally, consider a generic density function ρ(x|θ) dependent on one or several parameters Riemannian geometry of subspaces then shows that the metric on the subspace M is determined by where again the scale factor of four is purely conventional and we have written ∂ a = ∂/∂θ a . The which is more informative than the mere overlap distance cos −1 ( ξ(θ)|ξ(θ ) ).

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In the context of statistical mechanics the curvature of M associated with the Fisher-Rao metric is The geometric analysis of the parametric subspace of the real Hilbert space extends, mutatis mutandis, 148 to the complex domain-for example, to the complex Hilbert space of states in quantum mechanics.

149
There are, however, some modifications arising, which we shall discuss now. Consider first the case 150 of a parametric curve |ξ(θ) satisfying the normalisation condition ξ(θ)|ξ(θ) = 1, where ξ(θ)| now 151 denotes the Hermitian conjugate of |ξ(θ) . In the complex case the condition ∂ θ ξ(θ)|ξ(θ) = 0 does not 152 imply ξ (θ)|ξ(θ) = 0 owing to the phase factor, so we require a modified expression for the proper 'velocity' vector. The squared velocity (with a factor of four) is then given by The simplest situation of a curve |ξ(θ) that arises in quantum mechanics is the solution to the 155 Schrödinger equation with initial condition |ξ(0) satisfying ξ(0)|ξ(0) = 1, where the parameter θ represents time. A short 157 calculation then shows that the squared velocity is given by the energy uncertainty: which of course is merely the statement of the Anandan-Aharanov relation [31]. From the viewpoint of 159 inference theory we can think of a situation in which a quantum system, prepared in an initial state, is 160 made to evolve under the influence of the HamiltonianĤ. After a passage of time an experimentalist 161 performs a measurement in order to estimate how much time has elapsed since its initial preparation.

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The Cramér-Rao relation then asserts that the quadratic error of time estimation is bounded below by In other words, the metric tensor is given by where the brackets in the subscripts denote symmetrisation (which is just the real part of the expression 177 without the symmetrisation). In particular, for N = 1 we recover the expression in (10).
where for simplicity of notation we have omitted the θ-dependence ofĤ, E n , and |φ n . Assuming that 194 the eigenvalues ofĤ(θ) are nondegenerate we can use first-order perturbation theory to deduce that Substituting this in (16) we find that Observe that the skew-symmetric form obtained from the imaginary part of the expression (19), without Evidently,Û transports the state |φ n (θ) into |φ n (θ + dθ) . The generators of this evolution are then 213 given by the observables It is then a short exercise to show that the Fisher-Rao metric is just the covariance matrix for the 215 observablesX a [8].

216
Now if we letΘ a denote the unbiased estimator for the parameter θ a , then the two operatorsΘ a andX a 217 are conjugate to each other. In particular, from the Cramér-Rao inequality we find that the covariance 218 matrix ofΘ a is bounded below by the reciprocal of the Fisher-Rao metric. Hence the operator pair 219 (Θ a ,X a ) for each a satisfies a Heisenberg-like uncertainty relation. As an example, suppose that there 220 is a single control parameter θ in the Hamiltonian, and thatΘ is the unbiased estimator for θ, satisfying 221 φ n (θ)|Θ|φ n (θ) = θ. (In general,Θ will not be a self-adjoint operator.) Suppose, further, that λ −1X 222 is the self-adjoint operator generating the shift in the parameter θ so that e −iX /λ φ(θ) = φ(θ + ) for 223 1. Here, λ is a constant such thatX /λ is dimensionless. In this situation, parameter estimate for θ 224 is limited by the variance lower bound of the form: where by ∆Θ 2 we mean the variance ofΘ, and similarly for ∆X 2 . It also follows (setting λ = 1) that where we have writtenĤ = ∂ θĤ . We remark that (22) represents a new type of uncertainty relation in 227 quantum mechanics that is in principle verifiable in laboratory experiments.

228
The perturbation analysis indicated above can also be applied to obtain an expression for the curvature 229 of a curve associated with a one-parameter family of eigenstates |φ n (θ) of a parametric Hamiltonian 230Ĥ (θ). In the one-parameter case (18) reduces to Assuming thatĤ(θ) is nondegenerate, the second-order term in perturbation series gives 232 |φ n = 2 which shows that φ n |φ n = 0. In this case the expression for the intrinsic curvature becomes: Substitution of (24) and (25) in (26) then gives the expression for the curvature, which, in turn, can be Additionally, it will be convenient to introduce eigenstates of the adjoint matrixK † : 249K † |χ n =κ n |χ n and χ n |K = κ n χ n |.
At an exceptional point, however, the convention χ EP |φ EP = 1 breaks down for the following 261 reason. Suppose that the two eigenstates |φ k and |φ l 'meet' at |φ EP . Evidently, the biorthogonality 262 condition implies that χ l |φ k = 0 and χ k |φ k = 0, but χ l | and χ k | will both approach χ EP | so that 263 we have χ EP |φ EP = 0. This feature is often referred to as 'self-orthogonality' in the literature. To 264 complete the basis for the eigenspace belonging to the degenerate eigenstate one needs to introduce 265 associated eigenvactors, or so-called Jordan vectors. We will return to this issue in the discussion of exceptional points in the section to follow, but for now we assume that the states are away from 267 degeneracies.

268
Away from exceptional points, and based on the convention that χ n |φ n = 1, the overlap distance s 269 between the two states |ξ and |η is now given by the expression: In particular, if |η = |ξ + |dξ is a neighbouring state to |ξ , then expanding (32) and retaining terms 271 of quadratic order, we obtain the following form of the Fubini-Study line element We remark that an analogous expression for the metric appears in [47], however, (33)  (K + ∂ aK dθ a + · · · )(|φ n + |∂ a φ n dθ a + · · · ) = (κ n + ∂ a κ n dθ a + · · · )(|φ n + |∂ a φ n dθ a · · · ), (34) where we have omitted explicit θ dependencies. Equating the terms linear in dθ we find So far the result is identical to that for a Hermitian Hamiltonian. However, the lack of orthogonality of 286 the eigenstates prevents us from using the projectorΦ m = |φ m φ m | to further simplify the expression.

287
Nevertheless, if we multiplyΠ m = |φ m χ m | from the left and rearrange terms we find For n = m we are led to the expression (cf. [45]):
To obtain an expression for |∂ k φ n , in [39] the operator (K − κ n 1) −1 is applied from the left in (35).

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This approach, however, is problematic on account of the fact that (K − κ n 1) is degenerate and thus not 291 invertible. The result of [39] can nevertheless be justified if we make the assumption that the perturbation vector |∂ a φ n dθ a is orthogonal to the dual vector |χ n . With this assumption, which turns out to be the 293 correct one, for n = m we divide both sides of (36) by κ m − κ n and sum over m = n to obtain where we have made use of the condition χ m |dφ m = 0.

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However, there are important differences, including the fact that the perturbation is not orthogonal to 297 the state |φ n , i.e. φ n |∂ a φ n = 0, but rather χ n |∂ a φ n = 0. It follows that under this assumption the 298 perturbation will necessarily change the overall complex phase of the eigenstate. This is nevertheless 299 natural under the geometry of the state space formulated from (33).

300
The metric geometry of the parameter space can now be determined if we substitute (38) in (33): for the metric breaks down near degeneracies, and one has to consider higher-order perturbations.

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In the case of a complex Hamiltonian, the situation is more severe on account of the fact that the Hermitian or more generally complex Hamiltonians (we remark that properties of exceptional points of 319 higher order where more than two eigenstates coalesce can be quite intricate; see, e.g., [54,55] and that 334 χ J EP |φ J EP = 0. that the eigenvalues and eigenvectors can be expanded in a power series with half-integral exponents.
and hence that From (48) and (49)  We conclude by remarking that although in the foregoing material we have placed some emphasis on 358 perturbative analysis for the geometry surrounding exceptional points so as to obtain generic expressions 359 for the metric, if a model is specified, then typically there is no need for evoking the perturbative approach 360 since the metric can be computed exactly. As an example, take the 2 × 2 HamiltonianK =σ x − iγσ z .

361
This Hamiltonian is PT symmetric, and has real eigenvalues in the region γ 2 < 1 where the eigenstates 362 are also PT symmetric. Specifically, the eigenstates ofK andK † are given by where n 2 ± = (1 ∓ iγ/ 1 − γ 2 )/2. A straightforward calculation then shows that the information metric 364 associated with the curve, say, |φ + (γ) , is given by on account of the relations: The nonperturbative expression in (52) shows exactly how the metric diverges as one approaches the thus recovering the perturbative result (50) in leading order of . 371 More generally, any curve of the form |ψ(γ) = c + |φ + (γ) + c − |φ − (γ) with fixed coefficients c ± 372 in this system possesses the metric (52) and will exhibit a curvature singularity at γ = 1. In the region 373 γ 2 1, on the other hand, we have G 1, and thus estimation of the parameter γ becomes unfeasible.